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Chapter 6: Problem 53

Recent crime reports indicate that 3.1 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. a. Calculate the probability exactly four thefts occur in a minute. b. What is the probability there are no thefts in a minute? c. What is the probability there is at least one theft in a minute?

Short Answer

Expert verified
a) 0.168, b) 0.045, c) 0.955.

Step by step solution

01

Understanding Poisson Distribution

The Poisson probability distribution is used when we are interested in the number of times an event occurs in a fixed interval of time or space independent of the last event. Here, we know that the average rate (\(\lambda\)) of motor vehicle thefts per minute is 3.1.
02

Formula for Poisson Probability

The probability of observing \(k\) events in an interval is given by the formula:\[P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\]where \(\lambda = 3.1\) and \(e\) is the base of the natural logarithm, approximately 2.71828.
03

Calculate the Probability of Exactly Four Thefts (a)

Using \(\lambda = 3.1\) and \(k = 4\), substitute these values into the Poisson formula:\[P(X = 4) = \frac{e^{-3.1} \cdot 3.1^4}{4!}\]Calculate \(e^{-3.1} \approx 0.045049\) and \(3.1^4 = 92.3521\), and finally \(4! = 24\).Then, \(P(X = 4) = \frac{0.045049 \times 92.3521}{24} \approx 0.168031\).
04

Calculate the Probability of No Thefts (b)

For \(k = 0\), use the Poisson formula:\[P(X = 0) = \frac{e^{-3.1} \cdot 3.1^0}{0!}\]Since \(0! = 1\), the formula simplifies to:\[P(X = 0) = e^{-3.1} = 0.045049\]
05

Calculate the Probability of At Least One Theft (c)

The probability of at least one theft is the complement of the probability of no thefts:\[P(X \geq 1) = 1 - P(X = 0)\]So, \[P(X \geq 1) = 1 - 0.045049 = 0.954951\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability in mathematics measures how likely an event is to occur.
It is represented by the letter "P" and ranges from 0 (impossible event) to 1 (certain event).
In our exercise, we're dealing with probabilities around motor vehicle thefts occurring within a given time frame, specifically using the Poisson distribution.
This distribution is a powerful tool for calculating probabilities of events that occur randomly in a fixed interval, here expressed as a number of thefts per minute.
Whether calculating the exact probability of four thefts or no thefts, understanding probability helps us interpret real-world data meaningfully.
Event Occurrence
In statistics, an event is any outcome or a set of outcomes of a random experiment.
Here, the "event" we are observing is the number of motor vehicle thefts per minute.
Understanding event occurrence is crucial, as it allows us to quantify how frequently these events happen and predict future happenings.
The Poisson distribution is particularly suited for such scenarios, where events happen independently and infrequently over a fixed period.
This model helps in anticipating various event frequencies such as exactly four thefts, no thefts, or at least one theft happening within that minute, aiding in decision-making and strategy development.
Fixed Interval
A fixed interval is a specific time or space period in which observations are made.
For the purpose of the Poisson distribution, this is crucial, as it defines where and when we observe events.
In our problem, the fixed interval is one minute, during which motor vehicle thefts are counted.
By fixing the interval, we allow for a consistent medium to compare probabilities consistently.
This consistency is key when calculating probabilities for different numbers of event occurrences, such as exactly four thefts or zero thefts in one minute using the provided formulas.
Theft Statistics
Theft statistics provide quantitative data about occurrences of thefts, useful for analysis and plotting trends.
In our exercise, theft statistics reveal an average of 3.1 motor vehicle thefts per minute nationwide.
This average is essential for determining the Poisson parameter, \(\lambda\), which in turn is vital for any probability calculations involved in this distribution.
By understanding these statistics, organizations and government agencies can better allocate resources for crime prevention and law enforcement strategies.
Such statistics can help not only in predicting and reacting to crime but also in developing policies that reduce motor vehicle thefts effectively.

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